Polymers containing carboxyl or anhydride groups are of great importance for a large variety of applications. For example, polyacrylic or polymethacrylic acids having a low degree of crosslinking, copolymers of C.sub.2 -C.sub.4 -olefins with maleic anhydride and copolymers of C.sub.1 -C.sub.4 -alkyl vinyl ethers with maleic anhydride are used as thickeners in the textile printing, cosmetics and pharmaceuticals sectors. Copolymers of maleic anhydride with methyl vinyl ether, in the form of the monoester with a monohydric C.sub.1 -C.sub.4 -alcohol, are becoming increasingly important as binders for hair sprays. Optimum polymers for this purpose are medium molecular weight or high molecular weight polymers which are generally prepared by precipitation polymerization in solvents in which the monomers are soluble and the resulting polymer is insoluble. Particularly suitable solvents for the precipitation polymerization are those which have a low transfer constant in the polymerization, such as benzene, toluene and chlorohydrocarbons, eg. carbon tetrachloride, dichloroethane and trichloroethane. Benzene, dichloroethane and trichloroethane are usually used. Since these solvents are physiologically unacceptable, they have to be substantially removed from the polymers before the latter are used, particularly when they are employed in the cosmetics sector. To do this, the polymer suspensions are usually dried in a drier, such as a spray drier, a fluidized-bed drier, a paddle drier or a similar apparatus. Even after thorough drying at relatively high temperatures and under reduced pressure for a relatively long time, the residual content of physiologically unacceptable solvents in the polymer is usually from 1 to 3% by weight. For example, a polyacrylic acid which had a low degree of crosslinking and was polymerized in benzene still had a residual benzene content of 1.4% by weight after it had been heated for 24 hours at 80.degree. C. under 0.01 mbar. Products having such a high benzene content are unsuitable for industrial use.